The broad objective of the proposed research is to develop a novel theory and practical methods for accurate reconstruction of three-dimensional (3D) tomographic images in single-photon emission computed tomography (SPECT). We believe that the new concepts, perspectives, and techniques developed in the proposed research will have a significant impact on ongoing research addressing advanced tomographic reconstruction techniques throughout the medical imaging community. The proposed research will investigate and develop methods that compensate for the effects of physical factors t hat arise in 3D SPECT. In particular, we will focus on the development of novel methods that compensate for uniform attenuation and distance-dependent spatial resolution, that provide closed-form (i.e., non-iterative) mathematical solutions, and that control noise in an optimal sense in 3D SPECT. We will extend the closed-form methods that we develop to 3D Spect with variable attenuation. Results of ongoing research by other investigators on compensation for scatter in 3D SPECT will be incorporated into the proposed research as they become available. Our preliminary theoretical and numerical studies ina the area of the proposed research are highly promising and have sparked considerable interest in the field of medical imaging science, revealing a variety of fundamental concepts and perspectives that were unknown previously in both 2D and 3D SPECT and that enrich our understanding of the 3D SPECT image reconstruction task. The specific aims of this proposal are; (1) Development f closed-form methods for accurate estimation of the ideal sinogram, (2) Development of 3D generalized Tretiak-Metz approaches, (3) Extension of the proposed methods to 3D SPECT with variable attenuation, (4) Implementation of the proposed methods, (5) Implementation of the proposed methods, (5) Quantitative evaluation of the proposed methods.